Let $\phi: G \to \text{Aut}(V)$ be an irreducible representation of a finite group $G$, where in some basis for $V$, all matrices $\phi(g)$ have real entries.
For this basis, is it true that $\phi(g)$ is always diagonal? If so, how can I see that? Otherwise, how can I construct such a basis? In my book they assume its trivial, but I don't see this.
No it's not true. For example, the dihedral group of order $8$ has such a representation of degree $2$. For $g$ of order $4$, $\phi(g)$ has complex eigenvalues, so it cannot be diagonal.